Quasi-exceptional domains

Exceptional domains are domains on which there exists a positive harmonic function, zero on the boundary and such that the normal derivative on the boundary is constant. Recent results classify (under some mild additional assumptions) exceptional domains as belonging to either a certain one-parameter family of simply periodic domains or one of its scaling limits. We introduce quasi-exceptional domains by allowing the boundary values to be different constants on each boundary component. This relaxed definition retains the interesting property of being an arclength quadrature domain, and also preserves the connection to the hollow vortex problem in fluid dynamics. We give a partial classification of such domains in terms of certain Abelian differentials. We also provide a new two-parameter family of periodic quasi-exceptional domains. These examples generalize the hollow vortex array found by Baker, Saffman, and Sheffield (1976). A degeneration of regions of this family provides doubly-connected examples